$\mathbb{Z}_p$-torus actions on positively curved manifolds

TL;DR

This paper investigates the symmetry properties of positively curved manifolds with specific group actions, improving existing bounds and introducing new coding theory tools for analyzing these geometric structures.

Contribution

It provides improved symmetry-rank bounds for positively curved manifolds with $bZ_p^r$-actions and introduces finite-length coding bounds relevant to geometric group actions.

Findings

Established a new symmetry-rank bound for large dimensions

Improved bounds for small primes $3 \,\leq p \leq 19$

Derived finite-length coding bounds with asymptotic advantages

Abstract

In this article, we study closed, positively curved $n$-manifolds that admit an effective, isometric $\mathbb{Z}_p^r$-action with a fixed point, where $p$ is an odd prime. For all sufficiently large $n$, we obtain a symmetry-rank bound in Theorem A that improves the $3n/8$ bound of Fang and Rong and of Ghazawneh. We improve on this bound for small odd primes $3\leq p\leq 19$ in Theorem B. One of our main tools comes from the theory of error-correcting codes and is of independent interest: we derive a finite-length Plotkin bound and a finite-length Elias-Bassalygo bound for $q$-ary codes and show that the finite-length Plotkin bound is asymptotically sharper for all primes $p \ge 23$.